Average Error: 29.4 → 0.1
Time: 1.8m
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -17316.428519489094:\\ \;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 21980.127985226965:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{1 + x} \cdot x\right) \cdot \left(x - 1\right) - \left(\left(1 + x\right) \cdot \frac{1 + x}{x - 1}\right) \cdot \left(1 + x\right)}{\left(1 + x\right) \cdot \left(x - 1\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -17316.428519489094:\\
\;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\

\mathbf{elif}\;x \le 21980.127985226965:\\
\;\;\;\;\frac{\frac{\left(\frac{x}{1 + x} \cdot x\right) \cdot \left(x - 1\right) - \left(\left(1 + x\right) \cdot \frac{1 + x}{x - 1}\right) \cdot \left(1 + x\right)}{\left(1 + x\right) \cdot \left(x - 1\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\

\end{array}
double f(double x) {
        double r32521180 = x;
        double r32521181 = 1.0;
        double r32521182 = r32521180 + r32521181;
        double r32521183 = r32521180 / r32521182;
        double r32521184 = r32521180 - r32521181;
        double r32521185 = r32521182 / r32521184;
        double r32521186 = r32521183 - r32521185;
        return r32521186;
}
↓
double f(double x) {
        double r32521187 = x;
        double r32521188 = -17316.428519489094;
        bool r32521189 = r32521187 <= r32521188;
        double r32521190 = -16.0;
        double r32521191 = r32521187 * r32521187;
        double r32521192 = r32521191 * r32521187;
        double r32521193 = r32521190 / r32521192;
        double r32521194 = 6.0;
        double r32521195 = r32521194 / r32521187;
        double r32521196 = 5.0;
        double r32521197 = r32521196 / r32521191;
        double r32521198 = r32521195 + r32521197;
        double r32521199 = r32521193 - r32521198;
        double r32521200 = 1.0;
        double r32521201 = r32521200 + r32521187;
        double r32521202 = r32521187 - r32521200;
        double r32521203 = r32521201 / r32521202;
        double r32521204 = r32521187 / r32521201;
        double r32521205 = r32521203 + r32521204;
        double r32521206 = r32521199 / r32521205;
        double r32521207 = 21980.127985226965;
        bool r32521208 = r32521187 <= r32521207;
        double r32521209 = r32521204 * r32521187;
        double r32521210 = r32521209 * r32521202;
        double r32521211 = r32521201 * r32521203;
        double r32521212 = r32521211 * r32521201;
        double r32521213 = r32521210 - r32521212;
        double r32521214 = r32521201 * r32521202;
        double r32521215 = r32521213 / r32521214;
        double r32521216 = r32521215 / r32521205;
        double r32521217 = r32521208 ? r32521216 : r32521206;
        double r32521218 = r32521189 ? r32521206 : r32521217;
        return r32521218;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -17316.428519489094 or 21980.127985226965 < x

    1. Initial program 59.4

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied flip--59.4

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]

    4. Taylor expanded around -inf 0.3

      \[\leadsto \frac{\color{blue}{-\left(16 \cdot \frac{1}{{x}^{3}} + \left(5 \cdot \frac{1}{{x}^{2}} + 6 \cdot \frac{1}{x}\right)\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    5. Simplified0.0

      \[\leadsto \frac{\color{blue}{\frac{-16}{x \cdot \left(x \cdot x\right)} - \left(\frac{5}{x \cdot x} + \frac{6}{x}\right)}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    if -17316.428519489094 < x < 21980.127985226965

    1. Initial program 0.1

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
    2. Using strategy rm

    3. Applied flip--0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}}\]
    4. Using strategy rm

    5. Applied associate-*r/0.1

      \[\leadsto \frac{\frac{x}{x + 1} \cdot \frac{x}{x + 1} - \color{blue}{\frac{\frac{x + 1}{x - 1} \cdot \left(x + 1\right)}{x - 1}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    6. Applied associate-*l/0.1

      \[\leadsto \frac{\color{blue}{\frac{x \cdot \frac{x}{x + 1}}{x + 1}} - \frac{\frac{x + 1}{x - 1} \cdot \left(x + 1\right)}{x - 1}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]

    7. Applied frac-sub0.1

      \[\leadsto \frac{\color{blue}{\frac{\left(x \cdot \frac{x}{x + 1}\right) \cdot \left(x - 1\right) - \left(x + 1\right) \cdot \left(\frac{x + 1}{x - 1} \cdot \left(x + 1\right)\right)}{\left(x + 1\right) \cdot \left(x - 1\right)}}}{\frac{x}{x + 1} + \frac{x + 1}{x - 1}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -17316.428519489094:\\ \;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{elif}\;x \le 21980.127985226965:\\ \;\;\;\;\frac{\frac{\left(\frac{x}{1 + x} \cdot x\right) \cdot \left(x - 1\right) - \left(\left(1 + x\right) \cdot \frac{1 + x}{x - 1}\right) \cdot \left(1 + x\right)}{\left(1 + x\right) \cdot \left(x - 1\right)}}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-16}{\left(x \cdot x\right) \cdot x} - \left(\frac{6}{x} + \frac{5}{x \cdot x}\right)}{\frac{1 + x}{x - 1} + \frac{x}{1 + x}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))